Dedicated to the application of mathematical modeling and fieldwork to the study of the history of cartography...and perhaps a few philosophical reflections on the above...

Author Climbing in the Queyras, Summer 2013

Sunday, October 08, 2006

Is the 1516 Carta Marina a Portolan Chart?

The 1516 World Map by Martin Waldseemuller, known as the Carta Marina, looks very much like a Portolan Chart even though it is a printed map. Many scholars have surmised that the sources for the map were in fact Portolan Charts but no one has ever attempted any calculations that might allow conclusions beyond these speculations.

The above figure (click on figures to enlarge view) is the sheet from the 1516 Carta Marina that contains Europe and most of the Mediterranean. Portolan charts contain two analytic features that are very important in distingushing them from other maps of the period. First, the axis of the Mediterranean basin is deflected or rotated by between 5 and 11 degrees. This orientation shift most likely results from an orientation to magnetic and not true North. A.C. Mitchell's paper "Chapters in the history of terrestrial magnetism" (Terrestrial Magnetism and Atmospheric Electricity, XLII (1937) 241-280) is one of the earliest to suggest this and is worth reading.

In order to test the Carta Marina for rotation of the Mediterranean basin we used an affine transformation and a Hampel estimator to control the error distribution of the chosen landmark points. The figure below shows the sheet of the Carta Marina with a distortion grid calculated using M-estimators. M-estimators (see links) are part of a large family of statistical estimation functions (known as robust) that try to reduce the effect of outliers or points of large error on the overall tranformation calculations.

Hampel gives several answers to the question of when to apply Robust estimators:

There are two observations which when combined give an answer. Often in statistics one is using a parametric model implying a very limited set of probability distributions, such as the common model of normally distributed errors, or that of exponentially distributed observations. Classical (parametric) statistics derives results under the assumption that these models were strictly true (this is especially important in our cartographic applications where are chosen landmarks are rarely evenly distributed). However, apart from some simple discrete models perhaps, such models are never exactly true. We may try to distinguish three main reasons for the derivations: (i) rounding and grouping and other "local inaccuracies''; (ii) the occurrence of "gross errors'' such as blunders in measuring, wrong decimal points, errors in copying, inadvertent measurement of a member of a different population, or just "something went wrong''; (iii) the model may have been conceived only as an approximation anyway, e.g. by virtue of the central limit theorem.

One can imagine using these estimators as an application of Tobler's first law of geography, "everything is related to everything else, but near things are more related than distant things" (see Waldo Tobler's article in Economic Geography 46 234-40).

The application of the transformation to the European sheet yields a rotation of 7.6 degrees for the Mediterranean which is consistent with a Portolan source. Below is the bare grid displayed for clarity. The other feature that most Portolan's display is a lining up of the tip of Brittany with the location of Venice, showing both places on the same east-west line. The area of Brittany should be displaced by more than 3 degrees. The 1516 map shows a lining up of these two places consistent with most Portolans. The cause of this distortion most likely comes from an error in the interpretation of the data from the Atlantic coast, the prototype being measured in Catalan miles which are shorter than the Italian miles typically used in the Mediterranean.

Although none of this conclusively proves that Waldseemuller used Portolan charts as sources for the 1516 Carta Marina it a least implies it as a possibility and suggests a place to look for possible prototypes of this region..

Exploring the Ruins of Tikal...

My article in Alpinist Magazine 41....called Vericality: the other blank on the map....

when not climbing in the Alps, mountain biking through some jungle or looking for Roman ruins in North Africa, is a Specialist in Modern Cartography and Geographic Information Science in the Geography and Map Division of the Library of Congress. A Fellow of the Royal Geographical Society he is the recent recipient of a J. L Heiberg Research and Exploration Fellowship for his work on the physical remains of Roman Centuriation and in 2010 was awarded a J.S. Best Fellowship from the American Geographical Society.
While at the Library of Congress, much of his research has concentrated on the use of computer modeling in the analysis of Roman, Medieval and Renaissance maps.
He is also a lecturer on the history of cartography and and Geographic Information Sciences in the Graduate School for Advanced Studies at the Johns Hopkins University.

Kant in the Wilderness: Thoreau's Geographic Turn

.....click on Henry to read my new article just published by the Thoreau Society

History of GIS...my new article in ArcNews....

...click on above to read.....

Exploring the Mer de Glace...

...and its melting....

In the Land of Melting Glaciers...

...working on medieval land charters and using them to reconstruct early land ownership law....

David Kendall's article "Recovery of Structure From Fragmentary Information" first showed how to use statistical methods to reconstruct geographical layouts and maps from chartularies and medieval deeds....

Kendall's early explorations...

...making maps from fragmentary data...

Kendall working on statistical map reconstuction.....

...I try in my reconstructions to follow in his footsteps....

Recent Appearance on C-Span American History Channel

Click to view video....

Deep in the gorges of Provence....

...all Bruce Chatwin needed was a leather jacket, 20 bucks in his pocket and a notebook and the world was his.....words to live by....

Eric Shipton surveying in the Karakorum....

...if I could only write about it all like he did....

Recent and forthcoming publications....

...another new article...just published in the Portolan....

In Nietzsche's Shadow: Searching for the Remains of Roman Cartography in Southern France

My newest book just released....

..click on the cover to link to the New York Times Review above....

Around the world...some images from recent travels and mapping projects....

Searching for and mapping the megaliths on the island of Bornholm....

" 'Amateur' field geographers can speak with authority about the clarifying effects on the mind of direct physical danger in the real world and there exists a terrible antagonism between field geographers and armchair academics. Not only do those in their armchairs think and write junk, obfuscation, obscurantism, and endlessly convoluted self-referral to their literature in windowless libraries, they do not care about the human condition.”

--William Bunge, Geography is a Field SubjectArea, 1983

It is Bunge’s words that I remember when I am out in the field in places like Tunisia and Algeria.

Exploring theTomb of Cleopatra II, near Tipasa, Algeria

Summer, 2011

...the tomb as it looked in 1926....

...entrance to the tomb....

...the underground aqueducts and sewers of Tipasa, Algeria...

...nothing but questions... trying to map their extent....Summer 2011...

Heidegger Climbing in the Mountains near Davos....

For Peter Gould.... who showed me there was something in Heidegger that geographers could use...

"Indeed space is still one of the things that is constitutive for the world... ---Martin Heidegger

I had read text after text of Martin Heidegger under the tutelage off one of the finest scholars it has ever been my privilege to meet. The philosopher Joseph Kockelmans always worked with parallel German and English texts, and with the Greek text too if it were appropriate, and slowlyopened up the thinking that lay behind Heidegger's tenacious quest. If anyone had told me we would spend sixteen weeks on the forty-five pages of Heidegger's "Anaximander Fragment," I would have laughed, but at the age of forty seven, I began to read properly.---Peter Gould, Becoming a Geographer

...a must read for all geographers and cartographers....

Wittgenstein & Cartographic Theory

...slightly modified [added to] selections from the Tractatus...

We picture facts to ourselves
A picture [map] presents a situation in logical space, the existence and non-existence of states of affairs.
A picture [map] is a model of reality.
A picture [map] is a fact.
A pictorial [cartographic] form is the possibility that things are related to one another in the same way as elements of a picture [cartographic objects].That is how a picture [map] is attached to reality; it reaches right out to it. It is laid against reality like a measure.
Every picture [map] is at the same time a logical[mathematical][spatial] one. Not every picture is however a spatial one. [every map is].It is as impossible to represent in language anything that contradicts logic as it is impossible to represent by its coordinates a figure that contradicts the laws of space [projection], or to give coordinates of a point that does not exist. [void]

...Wittgenstein's grave at Acsension Parish Burial Ground, Cambridge....notice the little ladder that recalls the last few statements of the Tractatus....

...my theory of cartographic history...

The actual maps that exist are but a tiny subset of the theoretical maps that could exist. These real maps are products of a very small number of trajectories through cartographic space...each with its own unique place in this mathematical construction. Every real map is surrounded by a tiny cluster of real or unreal neighbors who are its ancestors and descendents...

...this notion takes its starting point from David Lewis' "On the Plurality of Worlds".

...think of maps as random graphs.....

...both of these books by Tim Robinson are must reads for any cartographer....or historian of maps.

....Robinson's reflections on cartography and the effect of landscape on abstract thought are among the most insightful in all of cartographic literature.....

Gian-Carlo Rota.....

His MIT lectures changed my way of thinking about mathematics....

His "End of Objectivity" lectures were groundbreaking and are currently not easy to come by.....

Galois Lattices....

....maps are represented using attribute tables and the connections between them are calculated using programs like Galacia....click on lattcies for more on Galacia......more details coming in a future full post...

Garrett Birkhoff....

...his Harvard Lectures on Lattice Theory...the class known as MATH 252...developed some of the most beautiful and applicable algebraic structures to the study of stemma and recensions.

The mathematics of the medieval Portolan chart

Click for the Washington Post Story on my modeling of Portolan Charts

Mathematical models of the 1507 Waldseemuller Map

...click to read the Washington Post article on my research....

This blog is featured at the National Library of Scotland's Georeferencing page

Waldseemuller and me....

Click on image for the video "Bridging the Gap Between Physical and Digital Preservation" about the encasing and digitization of the Waldseemuller map...

My Recent Books...

The Naming of America: Martin Waldseemuller's 1507 World Map and the Cosmographiae Introductio

...this is my translation and commentary on Waldseemuller's Cosmographiae Introductio....click on image to order from Amazon...

Reviews of The Naming of America

Imago Mundi:

....Hessler’s nuanced translation brings to life this dynamic period of cartographic history and the theories used by these early sixteenth-century cosmographers.His close attention to the Latin and his extensive notes reflect a level of serious scholarship that should place this book on the reading list for all graduate seminars focused on understanding the production of early modern knowledge. The presentation, graphic aesthetic and accessibility of the text will make this a favourite for general readers, as well, and should be on the wish list of everyone interested in early American history and cartography.

California Literary Review:

...It lurks in the background of our childhood imagination, now and again roaring back in adulthood to remind us of possibilities. A map of the world, that fixture in elementary classrooms, has always been a book masquerading as a flat piece of paper. Like layers of the earth for geologists, maps offer a glinting sample of the past. And when it comes to the Waldseemüller map, the Universalis Cosmographia that forms the subject of The Naming of America by John W. Hessler, there are earth-shattering discoveries to be found. Let it be said, up front, that The Naming of America is not a popular work in the vein of Doris Kearns Goodwin or Stephen Ambrose. Hessler’s is a scholarly affair, impeccably printed, where the footnotes are as long as the text, and controversies are discussed with dry impartiality....

Warntz and Tobler...two mathematical pioneers....

William Warntz on topological and cartographic surfaces....

We now look upon maps not only as stores for spatially ordered information, but also as a means for the graphical solution of certain spatial problems for which the mathematics proves to be intractable, and to produce the necessary spatial transformations for hypothesis testing....The modern geographer concieves of spatial structures and spatial processes as applying not only to such things as landforms....but also to social, economic, and cultural phenomena portraying not only conventional densities but other things such as field quantity potentials, probabilites, refractions etc etc. Always these conceptual patterns may be regarded as overlying the surface of the real earth and the geometrical and topological characteristics of these patterns, as tranformed mathematically or graphically, thus describe aspects of the geography of the real world...

---Spatial Order, Harvard Papers in Theoretical Geography 1, 1967

We recognize yet another role for maps. In the solution of certain problems for which the mathematics, however elegantly stated, is intractable, graphical solutions are possible. This is especially true with regard to "existence theorems". There are many cases in which the graphical solution to a spatial problem turns out to be a map in the full geographical sense of the term, "map." Thus a map is a solution to the problem.

The basic ideas of the theory of convex sets are naturally and easily appreciated when examined in a geographical context, rather than in a non-spatial one...in this respect geography is truly elegant.

Maps showing regional classification can be regarded as logic diagrams. Mapping of sets is a general mathematical concept. Geographical mapping is merely a special case of this.

Tobler's computer program, called Bi-dimensional Regression marks the real beginning of the mathematical analysis of historic maps. The paper was published in 1977 along with a Fortran program which performed an 'empirical transformation regressing an independent plane configuration against a similar configuration.' The program itself constructs this geometric transformation by estimating two sets of bivariate points on a square lattice that have been interpolated from the originally irregularly arranged original observations. This is very much analogous to modern landmark morphometrics using thin-plate splines and various other methods. The curvilinear regression coefficients are represented by Tobler as a spatially varying, but coordinate invariant, second-order tensor field...

Tobler's paper is geometrically insightful and, although being from the 1970's, still contains ideas that are yet to be fully explored in the study of the history of cartography.... Two-dimensional asymmetric tensor analysis for example... his work inspires everything I do....

By far the best theoretical work on the methodology of the history of cartography.

What the historian of cartography should be concerned with is a systematic study of the factors affecting error, and seek to establish their cause and variability and the statistical parameters by which error is characterized.....